Of course,this estimateis not nearlyas accurateas we wouldlike. For functionssuchas we
can easily find an antiderivative with which we can apply the Fundamental Theorem that
But it is not alwayseasyto find an antiderivative.Indeed,for manyintegralsit is
impossibleto find an antiderivative.Anotherissueconcernsthe questionsaboutthe accuracyof the approx-
imation.In particular, how largeshouldwe take n so that the TrapezoidalEstimatefor is accurate
to withina givenvalue,say? As with our LinearApproximationsin the Lessonon Approximation
Errors,we can statea methodthat ensuresour approximationto be withina specifiedvalue.
ErrorEstimatesfor Simpson'sRule
We wouldlike to haveconfidencein the approximationswe make.Hencewe can choose to ensurethat
the errorsare withinacceptableboundaries.The followingmethodillustrateshow we can choosea sufficiently
large
Suppose for Thenthe errorestimateis givenby
Example2:
Find so that the TrapezoidalEstimatefor is accurateto
Solution:
We needto find suchthat We startby notingthat for
Hencewe can take to find our errorbound.
We needto solvethe followinginequalityfor :